Analyzing Student Language Preferences and Calculating Ratios

The problem provides data on the language preferences of 100 students in a class. We are given the number of students who like specific languages, either exclusively or in combination. To solve the question about the ratio of students liking Spanish to those liking German, we first need to find the total number of students who like each of these languages based on the given information.

Understanding the Given Data

Let's break down the student numbers for each language preference category:

• Students who like only Spanish: 20
• Students who like only French: 30
• Students who like only German: 15
• Students who like both Spanish and French: 10 (Based on the phrasing and context with 'only' categories, this usually implies liking Spanish and French but not German, unless specified otherwise. We will assume this interpretation.)
• Students who like both German and French: 5 (Similarly, assuming liking German and French but not Spanish.)

The remaining students like only English. Let's calculate the number of students in the categories listed above:

\(20 + 30 + 15 + 10 + 5 = 80\) students.

Total students in the class are 100. So, the number of students who like only English is the remaining count:

\(100 - 80 = 20\) students.

Calculating Total Students Liking Spanish

Students who like Spanish include those who like:

• Only Spanish
• Spanish and French (but not German)
• Potentially Spanish and German, or Spanish, French, and German (if such categories existed and were not zero)

Based on the provided data, the categories that include liking Spanish are 'only Spanish' and 'both Spanish and French'. There is no mention of students liking Spanish and German, or all three languages. Therefore, the total number of students who like Spanish is the sum of these two groups:

Total students liking Spanish = (Students who like only Spanish) + (Students who like Spanish and French)

Total students liking Spanish = \(20 + 10 = 30\)

Calculating Total Students Liking German

Similarly, students who like German include those who like:

• Only German
• German and French (but not Spanish)
• Potentially Spanish and German, or Spanish, French, and German (if such categories existed and were not zero)

Based on the provided data, the categories that include liking German are 'only German' and 'both German and French'. There is no mention of students liking Spanish and German, or all three languages. Therefore, the total number of students who like German is the sum of these two groups:

Total students liking German = (Students who like only German) + (Students who like German and French)

Total students liking German = \(15 + 5 = 20\)

Calculating the Ratio

The question asks for the ratio of students who like Spanish to those who like German. This ratio is calculated as:

Ratio = \(\frac{\text{Total students liking Spanish}}{\text{Total students liking German}}\)

Ratio = \(\frac{30}{20}\)

Now, we simplify the ratio by dividing both the numerator and the denominator by their greatest common divisor, which is 10:

Ratio = \(\frac{30 \div 10}{20 \div 10} = \frac{3}{2}\)

The ratio of students who like Spanish to those who like German is 3/2.

Summary of Data

The ratio of students liking Spanish to students liking German is \(30 : 20\), which simplifies to \(3 : 2\) or \(\frac{3}{2}\).

Revision Table: Key Concepts

Additional Information: Language Preference Surveys

Problems like this are common in data analysis and quantitative reasoning. They often involve interpreting survey data where individuals can belong to multiple categories (e.g., liking more than one language). Using a Venn diagram can be a helpful visual tool to represent the different overlapping groups, although it wasn't strictly necessary for this specific question as there was no overlap between Spanish & German directly mentioned, or all three languages.

When solving such problems, always carefully read the wording, especially terms like "only" and "both," as they define specific, non-overlapping segments within the larger sets of people who like a particular item (like a language).